Existence and Uniqueness Results of Coupled Fractional-Order Differential Systems Involving Riemann–Liouville Derivative in the Space Wa+γ1,1(a,b)×Wa+γ2,1(a,b) with Perov’s Fixed Point Theorem

This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces E=Wa+γ1,1(a,b)×Wa+γ2,1(a,b). Our strategy is to endow the space E with a vector-valued norm and apply the Perov fixed point theorem. An example is given to show the usefulness of our main results.


Introduction
The beauty of fractional calculus lies in finding the derivative and integration of an operator for any order. Therefore, fractional derivatives became helpful in studying the anomalous behavior of dynamical systems in biology, viscoelasticity, bioengineering, electrochemistry, etc. [1][2][3][4]. So, the subject of fractional calculus has become the center of attractive research.
The origins of fractional calculus go back to the end of the 17th century, starting from the discussion between Leibniz and de l'Hôpital regarding the meaning of Moreover, several investigations have been introduced to develop and study this important mathematical field, including Liouville, Riemann, Abel, Riesz, Weyl, Hadamard, and Caputo.
To ensure a solution of some nonlinear problems, researchers utilize some suitable fixed point theorems. One of these theorems is the Banach contraction principle. Perov, in 1965, extended the Banach contraction principle to the vector-valued metric spaces by replacing the contraction factor with a matrix that converges to zero [6]. Perov's fixed point theorem is one of the crucial methods to prove an existence solution of systems of differential equations, fractional differential equations, and integral equations in N variables; see [7][8][9][10], and the references cited therein.
Recently, a number of interesting papers [11][12][13] on the solvability of mathematical problems in Sobolev spaces W n,p (R + ) [14] with the help of fixed point theory have been presented. In [15], the authors utilized the Riemann-Liouville derivative to introduce the left fractional Sobolev spaces W γ,p a + (a, b), where −∞ < a < b < +∞, 1 ≤ p < +∞, and n − 1 < γ ≤ n, n ∈ N.
Boucenna et al. [16] proved the existence of the solution for the following initial value problem: 0 + is the Riemann-Liouville fractional derivative of order α, 1 < α ≤ 2, RL I 2−α 0 + is the Riemann-Liouville fractional integral, and f satisfies some certain conditions.
Our work is devoted to studying the existence and the uniqueness of a coupled system of fractional differential equations of the form: in the generalized Banach space W We organize our paper as follows. In Section 2, we present the essential definitions and background that will be used in the rest of our paper. In Section 3, we use the Perov fixed point theorem to establish the existence and uniqueness of a solution of problem (1). In the last section, we give an example to show the applicability of our main result.

Preliminaries
In this section, we introduce some notations, definitions, and auxiliary results that will be used later. We begin with the following basic definitions of fractional calculus. Definition 1. The Riemann-Liouville fractional integral [17] of order γ for a function f ∈ L 1 (0, b) is defined as Definition 2. The Riemann-Liouville fractional derivative of order γ of a function f is given by where n = In(γ) + 1, and In(γ) denotes the integer part of γ [17,18].
The following assertion shows that fractional differentiation is an operation inverse to the fractional integration from the left.
Lemma 2. Let z, t ∈ R [20]. Then, On M m×n (R + ), we define a partial order relation as follows: Let M, N ∈ M m×n (R + ), m ≥ 1, and n ≥ 1. Put M = (M i,j ) 1≤j≤m 1≤i≤n and N = (N i,j ) 1≤j≤m 1≤i≤n . Then, Definition 4. Let E be a vector space on K = R or C. By a generalized norm on E , we mean a map · G : E −→ R n satisfying the following properties: G for all ∈ E and λ ∈ K, and (iii) is called a generalized Banach space (in short, GBS), if the vector metric space generated by its vector metric is complete.

Remark 1.
In the sense of Perov, the definitions of convergence sequence, continuity, and open and closed subsets in a GBS are similar to those for usual Banach spaces [21].

Definition 5.
Let (E , · G ) be a GBS and let K be a subset of E . Then, K is said to be G-bounded, if there exists a vector V ∈ R n + such that for all ∈ K, G V.

Definition 7.
Let (E , d G ) be a complete generalized metric space and let N be an operator from E into itself. N is called G-Lipschitzian if there exists a square matrix M of non-negative numbers such that If the matrix M converges to zero, then N is called an M-contraction.
The following result is due to Perov, which is a generalization of the Banach contraction principle. Theorem 1. Let E be a complete generalized metric space and let N : E −→ E be an M-contraction operator [6]. Then, N has a unique fixed point in E .
From [15], the left fractional Sobolev space of order γ is the set W γ,1 0 + (0, b) defined as follows: endowed with the norm .

Main Results
In this section, we study the existence and the uniqueness of a solution for a coupled system of fractional differential equations (1). (1) if and only if it is a solution of the following problem:

Lemma 5. is a solution of System
Hence, Reciprocally, by returning to (2), Hence, by replacing β i with α i − 1, we get (2) if and only if it is a solution of the following fractional integral equation system:

Lemma 6. is a solution of System
Proof. The proof of the above lemma can be found in [16].
In the rest of the paper, we assume the following hypotheses: and for each (t, 1 , 2 , 3 ), (t, x 1 , x 2 , x 3 ) ∈ (0, b) × R × R × R and for i ∈ {1, 2}, we have: (H 2 ) For each i = 1, 2, there is a positive number ξ i such that converges to zero, and there is r ∈ R 2 + that fulfills Proof. We define the operator N : W Step 1: First, we shall show that the mapping is well defined. Using our hypotheses, for arbitrarily fixed t ∈ (0, b), ∈ W β 1 ,1 Additionally, Thus, This means that the operator N maps W Keeping in mind that the vector r fulfills (5), we find that for all ∈B r and i = 1, 2, Due to (6), we derive that N is a mapping fromB r intoB r .
Step 2: Our claim here is to prove that the operator N is G-contractive. To this end, let On the other hand, for i = 1, 2, we have ). Then, This means that the operator N is G-contractive, and thus Perov fixed point Theorem 1 ensures that System (1) has a unique solution. Example 1. Consider the following initial value problem of a nonlinear coupled fractional derivative Then, sin(arctan(t)( 1 (t) + 2 (t))).